1917]
A SIMPLE PROBLEM IN FORCED LUBRICATION
515
to the axis of symmetry, and r is the distance of any point from that axis, Fig. 2. The velocities in the three directions are respectively u, v, w, and in virtue of the symmetry, they are all independent of 0. The motion is supposed to be steady, that is, the same at all times, and the inertia of the fluid is neglected. Under these conditions it is easy to recognize that w may be supposed to vanish throughout, and that v is given by
(1)
where z is measured from the fixed surface, so that v there vanishes.
In like manner the boundary conditions at z = 0 and z = h, as well as the equation of continuity, are satisfied by
u—C --- , ..............................(2)
•where G is a constant. The total flow U, representing the volume of lubricant fed in unit time, which flows past every cylindrical surface of radius r, is
U^ZTTT udz = ^-.........................(3)
JO O •
When the inertia terms are neglected, and attention is paid to the symmetry, the formal equations in cylindrical coordinates* are
Q=»V2,y_±. /K\
^.2 ' .................................\>JS
-£- = p^72w, .................................(6)
p denoting the pressure and p the viscosity, where
dr* r dr dz* *
* Basset's Hydrodynamics, Vol. n. p. 244, 1888.
33—2 In the original statement there was an. error, pointed out by Mr W. Pettingill.